Are the words "I, we, us, his, her, he, she" all prohibited in thesis writing?

These words are not necessarily prohibited, but there is an old norm in academic writing to avoid personal pronouns (the pronouns you listed). The reasoning behind the norm is that it makes for more objective writing, but it can also lead to the use of quite awkward passive voice phrasing. Because avoiding these pronouns does not necessarily make writing better, there is a counter-trend today which emphasizes writing clearly, even if that means you occasionally use "I" or "we".

Your supervisor will ultimately be one of the people evaluating your thesis, so it is important to take their preferences into account, but if you feel that writing without pronouns leads to too many awkward phrasings or otherwise makes your writing less clear, then I think it is worth pointing that out to your professor.

Note that this also tends to vary by discipline. In some fields, for example, the use of "we" to refer to the author (and collaborators or the readers) is entirely normal. In other fields, though, I have heard that it sounds pretentious. Try asking your colleagues and other mentors what they think the norms are in your field as well.

This is highly field dependent. Actually, in certain social fields such as women/gender studies, African American studies, ethnography, etc. it is required to use "I", to disclose any biases. "I am a 30 year old white male" etc.

I know advisers that would outright reject a thesis that doesn't explicitly use "I" in this manner (or at least something like "the author is ___").

The particulars vary incredibly by field and by journal. It's a fairly old practice to try and use passive form instead of active form, which appears to be what your instructor is suggesting. There is no "list" so much as the idea is to talk from the standpoint of what was being done (The experiment was conducted vs. I conducted the experiment). It has been suggested that the former passive form is harder to understand and the latter active form is preferred for clarity, but many academics (typically older professors, set in their ways) like the "traditional" passive style.